00000317.htm

300 X EXISTENCE THEOREMS

(b) Suppose in addition that E is finite dimensional; using the result of (a), give a new
proof of Peano's theorem (Section 10.5, Problem 4(b); use Ascoli's theorem (7.5.7) and
the Weierstrass approximation theorem (7.4.1)).

2. (a) In the polydisk P: |/ — t₀\ < a, |jc —je₀| < b in R², let g, h be two real valued
continuous functions such that g(t, x) < h(t₉ x) in P. Let u (resp. v) be a solution of
_x' == g(t_y x) (resp. xf = h(t_y x)) defined in an interval [/₀, t₀ + c[, taking its values in
]jc₀ — b,x₀ 4- b[ and such that «(/₀) = *o (resp. v(t₀) = x₀); show that u(t) < v(t)
foit₀<t<t₀ + c (consider the l.u.b. of the points s in [t₀, t₀ -f c[ such that«(/) < v(t)
for t₀<t< s).

(b) Let g be continuous and real valued in P, and let u be the maximal solution of
x' = g(t x) corresponding to (t₀,x_Q) (Section 10.5, Problem 7); suppose u is defined
(at least,) in an interval [t_Q, t₀ + c[ and takes its values in ]x₀ — b, X_Q + b[. Show
that in every compact interval [/₀ >t₀ + d] contained in [/₀, t₀ + c[, the maximal and
minimal solutions of x' = g(t_f x) + e are defined and take their values in ]JC_Q — 6, jc₀ + b[
as soon as e > 0 is small enough, and converge uniformly to u when e tends to 0.
(Given S_Q > 0, there exists an s > t₀ such that the maximal and minimal solutions of all
the equations x'=g(t, x) + e for 0 ^ e ^ s₀, corresponding to (t_Q, *o), are defined and
take their values in ]x₀ — b,x₀ + b[ for tin [t₀ ,s]; observe that all these functions form
an equicontinuous set in [t_Q, s], and prove the uniform convergence to u in [/₀, s] by
applying the result of (a), Ascoli's theorem (7.5.7), (10.4.3), and (8.7.8). Finally, show
that the l.u.b. of the numbers d .having the stated property is necessarily equal to c,
using in particular the last statement of Section 10.5, Problem 7.)

(c) In the polydisk P, let g and h be two continuous real valued functions such that
g(t_t x) ^ h(t, x) in P. Let [/₀, to + c] be an interval in which a solution u of x' = g(t_t x)
such that u(t₀) = x₀, and the maximal solution v of jc⁷ = h(t, x) corresponding to
(*o > x_Q) are defined and take their values in ]x₀ — b, x₀ + b[. Show that u(t) ^ v(t)
for f<> ^ t ^ t₀ + c (apply (a) and (b)).

3. (a) Show that the conclusions of Problem 6(a) of Section 10.5 are still valid when E is
finite dimensional, and the assumptions are modified as follows: (I)/is supposed to be
continuous in I x H (when K = R), but not necessarily locally lipschitzian; (2) <p is the
maximal solution (Section 10.5, Problem 7) of the equation z'~h(s,z) in [0, a],
corresponding to the point (0, 0). (Use the results of Problems l(a) and 2(b), and apply
the diagonal process as in Problem 4(b) of Section 10.5.)

(b) Suppose in addition that there exists a sequence (Y_n)_n>0 of real valued functions,
continuous in [0, a], taking their values in [0, r]₉ such that for n ^ 1, Y_n(s) =

[f, Y_n_i(£)) d£ for 0 < s ^ a. Let y₀ be continuous in J if K = R, analytic in J

if K = C, with values in S, and such that \\y₀(t) — x₀\\ ^ Y₀(|f— t_Q\) in J. Show
that there exists a sequence (y_n)_n^i of mappings of J into S, which are continuous if

K = R, analytic if K = C, and such that >>„(/) = *o-l- f(6,y_n-i(0))dQ, and that

\\y_n(t) — JC_Q || ^ Y_n(|/ — t₀\) in J for every n ^ 1. When K = C, conclude that the se-
quence (y_n) converges in J (uniformly in every compact subset of J) to the unique
solution u of x' =/(f, x). (Use (9.13.2) and the proof of (10.4.5).) Is this last statement
still true when K = R and/ is not supposed to be locally lipschitzian (cf. Section 10.5,
Problem 2)?

(a) Let I = fro, f₀ + c[ <= R, and let co be a real valued continuous
function ^ 0, defined in I x R. Let S be an open ball of center x₀ in E, and let
/ be a continuous mapping of I x S into E such that for / e I, x_t E S and x₂ e S,
\\f(t,xi) — f(t,x₂)\\ ^ a>(t, \\xi — x₂11). Let u, v be two solutions of x' =f(t,x), definedin



	300 X EXISTENCE THEOREMS (b) Suppose in addition that E is finite dimensional; using the result of (a), give a new proof of Peano's theorem (Section 10.5, Problem 4(b); use Ascoli's theorem (7.5.7) and the Weierstrass approximation theorem (7.4.1)). 2. (a) In the polydisk P: \|/ — t₀\ < a, \|jc —je₀\| < b in R², let g, h be two real valued continuous functions such that g(t, x) < h(t₉ x) in P. Let u (resp. v) be a solution of _x' == g(t_y x) (resp. xf = h(t_y x)) defined in an interval [/₀, t₀ + c[, taking its values in ]jc₀ — b,x₀ 4- b[ and such that «(/₀) = o (resp. v(t₀) = x₀);* show that u(t) < v(t) foit₀<t<t₀ + c (consider the l.u.b. of the points s in [t₀, t₀ -f c[ such that«(/) < v(t) for t₀<t< s). (b) Let g be continuous and real valued in P, and let u be the maximal solution of x' = g(t x) corresponding to (t₀,x_Q) (Section 10.5, Problem 7); suppose u is defined (at least,) in an interval [t_Q, t₀ + c[ and takes its values in ]x₀ — b, X_Q + b[. Show that in every compact interval [/₀ >t₀ + d] contained in [/₀, t₀ + c[, the maximal and minimal solutions of x' = g(t_f x) + e are defined and take their values in ]JC_Q — 6, jc₀ + b[ as soon as e > 0 is small enough, and converge uniformly to u when e tends to 0. (Given S_Q > 0, there exists an s > t₀ such that the maximal and minimal solutions of all the equations x'=g(t, x) + e for 0 ^ e ^ s₀, corresponding to (t_Q, o), are defined and take their values in ]x₀ — b,x₀ + b[ for tin [t₀ ,s];* observe that all these functions form an equicontinuous set in [t_Q, s], and prove the uniform convergence to u in [/₀, s] by applying the result of (a), Ascoli's theorem (7.5.7), (10.4.3), and (8.7.8). Finally, show that the l.u.b. of the numbers d .having the stated property is necessarily equal to c, using in particular the last statement of Section 10.5, Problem 7.) (c) In the polydisk P, let g and h be two continuous real valued functions such that g(t_t x) ^ h(t, x) in P. Let [/₀, to + c] be an interval in which a solution u of x' = g(t_t x) such that u(t₀) = x₀, and the maximal solution v of jc⁷ = h(t, x) corresponding to (o > x_Q)* are defined and take their values in ]x₀ — b, x₀ + b[. Show that u(t) ^ v(t) for f<> ^ t ^ t₀ + c (apply (a) and (b)). 3. (a) Show that the conclusions of Problem 6(a) of Section 10.5 are still valid when E is finite dimensional, and the assumptions are modified as follows: (I)/is supposed to be continuous in I x H (when K = R), but not necessarily locally lipschitzian; (2) <p is the maximal solution (Section 10.5, Problem 7) of the equation z'~h(s,z) in [0, a], corresponding to the point (0, 0). (Use the results of Problems l(a) and 2(b), and apply the diagonal process as in Problem 4(b) of Section 10.5.) (b) Suppose in addition that there exists a sequence (Y_n)_n>0 of real valued functions, continuous in [0, a], taking their values in [0, r]₉ such that for n ^ 1, Y_n(s) = [f, Y_n_i(£)) d£ for 0 < s ^ a. Let y₀ be continuous in J if K = R, analytic in J if K = C, with values in S, and such that \\y₀(t) — x₀\\ ^ Y₀(\|f— t_Q\) in J. Show that there exists a sequence (y_n)_n^i of mappings of J into S, which are continuous if

	K = R, analytic if K = C, and such that >>„(/) = o-l- f(6,y_n-i(0))dQ,* and that Jo \\y_n(t) — JC_Q \|\| ^ Y_n(\|/ — t₀\) in J for every n ^ 1. When K = C, conclude that the se- quence (y_n) converges in J (uniformly in every compact subset of J) to the unique solution u of x' =/(f, x). (Use (9.13.2) and the proof of (10.4.5).) Is this last statement still true when K = R and/ is not supposed to be locally lipschitzian (cf. Section 10.5, Problem 2)? (a) Let I = fro, f₀ + c[ <= R, and let co be a real valued continuous function ^ 0, defined in I x R. Let S be an open ball of center x₀ in E, and let / be a continuous mapping of I x S into E such that for / e I, x_t E S and x₂ e S, \\f(t,xi) — f(t,x₂)\\ ^ a>(t, \\xi — x₂11). Let u, v be two solutions of x' =f(t,x), definedin